How to Texture Paint on an Object using Local Symmetry of Another Intersecting Object?
I saw a Kent Trammel video on youtube that partially gave clues to how this might be possible, so I set up a basic scene in 2.8 to try my hand at applying the sculpting workaround to Texture Paint. You can see the normal symmetry painting there in the first screen.

I then turn on Face select mask mode and select the faces I want to paint on.

I go to edit mode and shift-s snap cursor to selection(this is where we will bring in another object as a helper)

I then scale the Plane object down once we have it in place - the local coordinates of this object are what we will use to get our symmetry set up.
I modified the plane in edit mode to extrude after creating the two loopcuts - these will be a cross that will help see the center of symmetry.

Select your paint object and then shift-select your helper object(the plane) and join them(ctrl-j)You will lose subdivision modifier, and will need to add it back before painting.

In edit mode, select the edge of the helper mesh and select linked(L) to select all of it, and use separate by selection.

Set the Helper object to Wire draw type in the Object Properties, and you can also turn off ability to render it in the Outliner.

Select your paint object and return to Texture Paint mode (Face Select Masking on to avoid painting on the opposite side of the object) and paint your desired design - here i test with just a cartoon face, as if Suzanne has a poorly made tattoo on her face like a rap artist. Painting one side of the cross hairs gets me the symmetrical result on the other side based on the helper object's local orientation.

After you are done, you can turn off Face Select masking and go back to Object mode and apply Location, Rotation and Scale to return the object origin to center of the paint object.